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January & February  2009, 23(1&2): 541-560. doi: 10.3934/dcds.2009.23.541

On the connection formulas of the third Painlevé transcendent

Roderick S. C. Wong 1, and  H. Y. Zhang 2,

1. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
2. 

Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Received  November 2007 Revised  February 2008 Published  September 2008

We consider the connection problem for the sine-Gordon PIII equation $u_{x x}+\frac{1}{x}u_{x}+\sin u=0,$ which is the most commonly studied case among all general third Painlevé transcendents. The connection formulas are derived by the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech. Anal., 1998).
Keywords: Connection formulas, sine-Gordon PIII equation, the third Painlevé transcendent (PIII), parabolic cylinder functions, Bessel functions, uniform asymptotics.
Mathematics Subject Classification: Primary: 33E17, 34M55; Secondary: 35Q53.
Citation: Roderick S. C. Wong, H. Y. Zhang. On the connection formulas of the third Painlevé transcendent. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 541-560. doi: 10.3934/dcds.2009.23.541
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